rules transformation rotation in geometry

Reflection over y-axis: T(x, y) = (- x, y) Reflection over x-axis: T( x, y) = ( x, - y) In other words, the line of reflection is directly in the middle of both points.Įxamples of transformation geometry in the coordinate plane. The line of reflection is equidistant from both red points, blue points, and green points. Notice the colored vertices for each of the triangles. Let's look at two very common reflections: a horizontal reflection and a vertical reflection. The transformation for this example would be T( x, y) = ( x+5, y+3).Ī reflection is a 'flip' of an object over a line.

More advanced transformation geometry is done on the coordinate plane. In this case, the rule is '5 to the right and 3 up.' You can also translate a pre-image to the left, down, or any combination of two of the four directions. The formal definition of a translation is 'every point of the pre-image is moved the same distance in the same direction to form the image.' Take a look at the picture below for some clarification.Įach translation follows a rule. The most basic transformation is the translation. Translations - Each Point is Moved the Same Way Negative factors reverse the direction of a segment to opposite side from the center.The original figure is called the pre-image the new (copied) picture is called the image of the transformation.Ī rigid transformation is one in which the pre-image and the image both have the exact same size and shape. Factors greater than 1 will stretch the segment, factors from 0 to 1 are shrinking this segment. Once chosen, to construct an image of any point on a plane as a result of this transformation, we have to connect a center of scaling with our point and stretch or shrink this segment by a scaling factor, leaving the center of scaling in place. You need to choose two parameters - (a) center of scaling and (b) factor of scaling. Once chosen, to construct an image of any point on a plane as a result of this transformation, we have to drop a perpendicular from our point onto an axis of reflection and extend it to the other side of the plane beyond this axis by the same distance.

You need to choose only one parameter - the axis (or line) of reflection. Once chosen, to construct an image of any point on a plane as a result of this transformation, we have to connect a center of rotation by a vector with our point and then rotate this vector around a center of rotation by an angle congruent to a chosen angle of rotation. You need to choose two parameters: (a) center of rotation - a fixed point on a plane and (b) angle of rotation. Once chosen, to construct an image of any point on a plane as a result of this transformation, we have to draw a line from this point parallel to a vector of translation and, in the same direction as chosen on the vector, move a point along this line by a chosen length.

These two parameters can be combined in one concept of a vector. You need to choose two parameters: (a) direction of the translation (straight line with a chosen direction) and (b) length of the shift (scalar).

Rules transformation rotation in geometry